Statements (24)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:King
|
| gptkbp:characterizedBy |
Descending chain condition on ideals
|
| gptkbp:contrastsWith |
Noetherian ring
|
| gptkbp:defines |
A ring in which the descending chain condition on ideals holds
|
| gptkbp:example |
Matrix rings over a division ring of finite dimension
Finite rings are Artinian |
| gptkbp:hasProperty |
Every simple module is finite-dimensional
Has only finitely many maximal ideals Jacobson radical is nilpotent |
| https://www.w3.org/2000/01/rdf-schema#label |
Artinian ring
|
| gptkbp:implies |
gptkb:Noetherian_ring_(in_the_commutative_case)
|
| gptkbp:introduced |
gptkb:Emil_Artin
|
| gptkbp:namedAfter |
gptkb:Emil_Artin
|
| gptkbp:property |
Every Artinian ring has finite length as a module over itself
Every non-empty set of ideals has a minimal element Every Artinian ring is semisimple if it is also a simple ring Every Artinian ring is Noetherian if it is also commutative Every Artinian ring is Noetherian if it is a left or right Artinian ring with the descending chain condition on left or right ideals |
| gptkbp:usedIn |
gptkb:algebra
module theory ring theory |
| gptkbp:bfsParent |
gptkb:commutative_algebra
gptkb:noncommutative_geometry |
| gptkbp:bfsLayer |
5
|